For Clement Attlee, it was the creation of the welfare state. For Anthony Eden, it was Suez. For Ted Heath, entry into Europe. For Jim Callaghan, the winter of discontent. For Tony Blair, Iraq.
It's early days for David Cameron. But what he most wants to avoid – apart from losing the next election - is for his government to be remembered solely for the cuts.
He wants it to go down in the history books for something else entirely – for reforming the welfare state that Attlee created and for mending the society that he claims has subsequently been broken.
It is no mean ambition. Welfare reform has proved to be a 'mission impossible' for successive Prime Ministers – even ones who told their welfare reform minsters to "think the unthinkable."
Mr Blair's failure to bring about meaningful change in this area – despite a 179-seat majority and a favourable economic headwind – has become symbolic of the many missed opportunities and thwarted hopes of his long premiership.
By contrast, Mr Cameron must depend for his parliamentary majority on the Liberal Democrats, while the economic environment could hardly be less conducive to his aim of moving people off welfare and into work.
Yet, perhaps by force of circumstances in the shape of the need to reduce the deficit, his government has embarked on a programme of change which, if successful, would amount to the biggest recasting of the welfare state since its inception.
Perhaps even more improbably, the reform programme is being overseen by Iain Duncan Smith, the quiet man who failed to turn up the volume and seemed destined to go down as no more than a footnote in Tory Party history.
Forced into the political wilderness in 2003, he skilfully reinvented himself as a Beveridge de nos jours, and now, as Work and Pensions Secretary, has the chance to put his radical ideas into practice.
At the heart of the changes announced in his White Paper this week are two relatively straightforward principles.
First, the replacement of the labyrinthine system of work-related state support with a single Universal Credit, and second, the idea that it will always pay better to work than remain on benefits.
So will it succeed where other attempts have failed? Well, in its favour is the fact that there is an unusual degree of political consensus over the central objectives of the changes.
"If the government gets this right we will support them because we accept the underlying principle of simplifying the benefits system and providing real incentives to work," said Shadow Work and Pensions Secretary Douglas Alexander.
New Labour leader Ed Miliband has already made clear he will not oppose the coalition for opposition's sake, and this is a wise strategic move on his part.
He realises there is a public consensus not just that the deficit needs to be cut, but that the dependency culture which has become entrenched in some deprived communities needs to be addressed.
But Labour's caveat, of course, is that the crackdown on benefits must go hand-in-hand with pro-growth policies to ensure the jobs are there for people to move into.
This highlights the biggest obstacle to Mr Duncan Smith's proposals – the fact that the government's cutbacks in other areas are likely to lead over the coming year to rising unemployment.
If there are not the jobs to go round, moving people off welfare into work becomes not just politically impossible but practically impossible.
As with much else, the fate of the government's welfare reform gamble depends on whether its greater economic policy gamble succeeds or fails.
